What is the specific term for this kind of pun?
In Shakespeare's Richard III Glouchester says:
Now is the winter of our discontent
Made glorious summer by this sun of York;
Study.com refers to this line with particular emphasis on the word sun:
We can also ascertain from the first few lines that Richard is celebrating his family's victory. Richard's brother, Edward IV (they are the sons of the Duke of York, so in the second line 'son' is actually a pun for sun) has taken the English crown from Henry IV.
I understand that this can be referred to as a paronomasia--a synonym of pun, but I have a vague recollection that this is a specific kind of pun where a word is changed by a letter or two.
The more specific paragram refers to the letters:
: a pun made by changing the letters of a word, especially the initial letter
Origin of PARAGRAM
Greek (skōmma) para gramma (joke) by letter, from para beside, beyond, by + gramma letter — more at para-, gram
M-W Emphasis mine
The general word paronomasia refers to the words:
Noun
- a play on words; a pun.
Origin
late 16th century: via Latin from Greek paronomasia, from para- ‘beside’ (expressing alteration) + onomasia ‘naming’ (from onomazein ‘to name,’ from onoma ‘a name’).
ODO Emphasis mine
Swine Lake, by James Marshall, is a well known example of changing one letter and adding another to Tchaikovsky's Swan Lake for dramatic effect.
According to Wikipedia, any pun is called a paranomasia. The particular type of pun you are referring to is:
The homophonic pun, a common type, uses word pairs which sound alike (homophones) but are not synonymous. Walter Redfern exemplified this type with his statement, "To pun is to treat homonyms as synonyms." For example, in George Carlin's phrase "Atheism is a non-prophet institution", the word "prophet" is put in place of its homophone "profit", altering the common phrase "non-profit institution". Similarly, the joke "Question: Why do we still have troops in Germany? Answer: To keep the Russians in Czech" relies on the aural ambiguity of the homophones "check" and "Czech". Often, puns are not strictly homophonic, but play on words of similar, not identical, sound as in the example from the "Pinky and the Brain" cartoon film series: "I think so, Brain, but if we give peas a chance, won't the lima beans feel left out?" which plays with the similar—but not identical—sound of "peas" and "peace".
I looked up ScotM's word, paragram, and found an interesting article on About.com.