Noddy is
"Knave Noddy" the obvious ancestor of Cribbage and its more elaborate relative Costly Colours. The earliest reference to it in the Oxford English Dictionary dates from 1589, though the basic term noddy, meaning a fool or simpleton, is nicely illustrated in the line "O beastly nody without brayne" (1550). In the gaming sense, Noddy is the name given to the Knave of the suit turned up at the start of play. Throughout the history of card games people have always tended to attach a personal name to the Knave of the best or trump suit. It was called Karnöffel in the 15th century game of that name, Pam (short for Pamphilus) in the 18th century game of Loo, and, most effectively of all - since it came to oust Knave itself - "Jack" in the 17th-century game of All Fours.
Here are some more historic and literary references:
Let not me take you at noddy anie more, least I present you to the parish for a gamster.
Thomas Nashe (attrib), An Almond for a Parrat (1589)
She'll sit up till you come, because she'll have you play a game at noddy.
Middleton Blurt, Master-Constable (1602)
By plaieing to much at primeroe and noddy he lost / Time and his monie to[o].
J. Day, Peregrinatio Scholastica (1610)
Noddie turn'd up, all made, yet lose the tricke.
Vadianus' Panegyric Verses in Coryat Crudities (1611)
You want four, and I two, and my deal: Now, knave noddy-no, hearts be trumps.
Samuel Foote, The Author (1757)
We have also... a game called noddy, the same, I believe, which we call niddy-noddy; another name of which is the Lord Mayor of Coventry.
Edward Moor: Suffolk Words and Phrases. (1823)
A sketchy description of the game by Randle Holme is interesting for some otherwise unrecorded scoring features and terminology. The whole thing reads as follows. (Note: The long S's will come out as question marks if your browser doesn't support UTF-8)
2 or 4 may play at it, 61 being up. Each perſon hath 3 cards and one turned up to which he makes as many caſts as he can. They are thus merkett, Flat back or King of Spads is ſix, Countenance or Queen of Hearts, four, Knave of the trump, 2, Knave of Hearts 5, a pair 4, pair Riall 12, a pair Taunt 24. Every 15 as you can make is 2, and every 25 is 2. In playing down the cards you have the ſame advantage of 15, 25, paires &c. and the next to 31 hath 1 caſt, if he make 31, there is 2 caſts.
(Merkett = marked, scored, pegged. Pur Taunte, also recorded as purtaunte, is a double pair royal (four of a kind).) In a section on terminology, Holme records "Roger" as a name assigned to the Knave of Hearts. Amongst other things, it was a generic name for a beggar posing as an impoverished university student.
HOW TO PLAY NODDY Based chiefly on Willughby's Book of Plaies (c.1665)
Preliminaries.
Noddy is played by two with a 52-card pack running A2345678910JQK in each suit. Cards count Ace 1, Two 2, and so on up to Ten 10, courts 10 each. Ace is always low both in cutting for deal and in play. It always counts 1, never 11, and whereas A23... is a valid run or sequence, AKQ... never is. A game is played up to 31 points over as many deals as it takes, and scores are recorded by pegging on a 31-hole Noddy board. (But they can easily be scored on a Cribbage board or writing.)
Deal.
Whoever cuts the lower card deals first and each then deals in turn. Deal three cards each in ones. Stack the rest face down and turn the top card for "trump" (so called). If it is a Jack, non-dealer pegs 2 for "Knave Noddy".
Object.
To peg points for making combinations both in the hand and in the play up to 31. The scoring features are as follows:
Knave Noddy
Jack of the trump suit: 1 (or 2 to non-dealer if turned up)
Point-counts
Fifteen (two or more cards totalling 15): 2
Twenty-five (three or more cards totalling 25): 1 per constituent card.
Thirty-one, or 'Hitter' (four or more cards totalling 31): 1 per constituent card.
Pairs
Pair (two cards of the same rank): 2
Pair royal, or prial (three of the same rank): 6
Double pair royal (all four of a rank): 12
Runs (Sequences)
Run of three: 2
Run of four: 4
Run of 5 or more: 1 per constituent card.
Flushes
Three or more cards of the same suit: 1 per constituent card.
Announcements.
Each in turn, starting with non-dealer, announces (but does not show) what scoring features they hold in the cards dealt, including the turn-up as if it were the fourth card of their hand, and peg the appropriate amount as they go along. As in Cribbage, any individual card can be counted more than once provided that it forms part of a distinct combination each time. (For example, 9-10-10-J counts as a pair of Tens and as two runs of three for a total of pegging of 6 holes.)
Play.
Non-dealer plays a card to the table and announces its face value. Each in turn thereafter plays another card and announces the combined total of all cards so far played. Whenever you play a card that creates a scoring feature in conjunction with the immediately preceding card or cards, you peg the appropriate score. The running total may not exceed 31. If you can't play without busting, you announce this fact (perhaps by saying 'Go', as in Crib), and your opponent may then keep playing and pegging as far as they can go without busting. For playing the last card of the series you peg 1 for the Go, or 2 for the "Hitter" (= the card that makes it 31 exactly). Any cards left unplayed in either player's hand are irrelevant, since (as in five-card Crib) there is no second round of play up to 31.
Example of play.
Bertha deals -
to Arthur, to herself, and turns up Announcements: Arthur calls "Fifteen 2 and a run of four for 6".
Bertha calls "Fifteen 2, fifteen 4, twenty-five 7, and a pair's 9".
Play: Arthur plays Four making 4, Bertha plays Nine making 13;
Arthur plays Six making 19, Bertha plays Six making 25 and pegging 6 (2 for the pair, 4 for the twenty-five);
Arthur plays Five making 30 and pegs 1 for last.
Score: On this deal Arthur has pegged 7 to Bertha's 15.
HOW TO PLAY NODDY
to Arthur,
to herself, and turns up 
Announcements: Arthur calls "Fifteen 2 and a run of four for 6".
