Rodrica does sound cute but I can't find any reference to it on Google as a real name.
There's Rodica (Slavic) and Roderica (German) which might fit the bill for you.
Otherwise:
Names with a d sound in the middle that could create a 'dy' derivative:
Lydia (Lyddie)
Hedda (Heddy)
Madeleine (Maddy)
Madison (Maddy)
Nadine (Naddy)
Audrey (doesn't really shorten)
Sydney (Syddie - ish! Bit odd!)
Andrea (Andy)
India (Indie)
Addison (Addy)
Names with a d sound in the middle that are already short so more like Roddy than Roderick:
Jody
Sadie
Other R names with a slightly unisex feel:
Rowan
Robyn
Riley
Reagan
Raven
Other R names with 3 syllables like Roderick that can also have derivatives:
Romilly (Romy)
Ramona (Mona but you probably wouldn't want that!)
Rebecca (Rivkah, Becca, Becky)
Rhiannon (Rhi, Anna)
Rowena (Rosie)
Rosalind (Rosie, Lina)
Rosanna (Rosa, Anna)
Other names from the same sort of era as Roderick that are quite strong sounding but have derivatives:
Annette (Annie)
Belinda (Lindy)
Clarice (Clary)
Deborah (Debby)
Evelyn (Evie)
Frances (Frankie)
Geraldine (Gerri)
Hester (probably doesn't shorten)
Irene (Reenie - maybe!)
Judith (Judy)
Katherine (Katy, Kitty)
Lillian (Lily)
Margaret (Maggie, Peggy)
Pauline (probably doesn't shorten)
Susan (Suzy, Sue)
Vivienne (Vivi)
Winifred (Winnie)